Does anyone know where I can get the recently shown current affairs
comedy series 'Don't Watch That, Watch This' on DVD?  It was shown on
the BBC over the past couple of months I think.

It does not seem to be available on DVD at this time, and I can find
no evidence of plans to release it.  You might continue to check
http://www.amazon.co.uk periodically to see if it is a planned
release, or perhaps another site in the UK similar to the US site
http://www.videoeta.com that discusses future DVD releases. (Again,
the videoeta site seems mostly geared for US releases, so you will
have to find a UK equivalent).